Today is the obitus of a pope. S Soter, Pope around 170ish, sent alms to the Corinthian Church during a famine; S Denys, Bishop of Corinth, wrote to his Brother about how this was the custom, ethos, of the Roman Church from the beginning, ap' arkhes, arkhethen. S Denys reveals that S Soter's Letter has been read at the Sunday Synaxis in Corinth and will continue to be read frequently, as the earlier Letter from S Clement still continued to be read.
Here we have got to the very heart of the question of Christian Unity and of the role played in that by the Roman Church. Communio, Koinonia, from the New Testament onwards, was expressed by hospitality (the receiving of fellow Christians from other Churches); by the sending of material assistance; and by the exchange of letters. If you wish to follow this up, try the roots koinon-/sunkoinon-, xen-, sustat-, and dekh-/dokh- in your Moulton. The root agap- perhaps sums it all up.
This is not all just a matter of kindliness and chumminess and having an affectionate sentiment beneath one's navel. It is shot through by dogma. When S Paul made his Gentile Christians collect money to be taken physically by their representatives to the Jerusalem Church, his motives were steely rather than soppy. By accepting the material assistance of their Gentile fellow Christians, the Jewish Christian headquarters would, in a very visible and tangible way, be accepting the validity of the Christian vocation of non-Jews. And in the early centuries we find in the giving or witholding of hospitality, and in the sending of letters, the diagnostics of what we would call 'being in communion'.
When S Ignatius refers to the Roman Church as "presiding over the agape", he is not paying some sort of conventional compliment to its soft-hearted generosity. He is alluding to the crucial, the nodal, centrality of the Roman Church in the links of Communio which manifest all the particular, local Churches, to be one Universal Church, Christ's Body.